The weather in Zanzibar largely depends on location. The sea surrounds the Island, and the wind and heat make for monsoon rains. The wet (or rainy) season in Zanzibar has two periods—the long and short showers.
From March till May (sometimes to June) is the long rain monsoon season. During this period, the year's volume of rain falls in Zanzibar, making the weather exceptionally humid. It's better to avoid Zanzibar during this period as the showers can last for days, though the sun may come out for a few hours at a time. As for the prices of accommodation, they are drastically cheaper!
The period of short rains called "mwaka" runs from November to December. The showers throughout these months are usually short and sharp. The sun takes over them as soon as possible. The temperature can reach up to 30°C with high humidity during this time.
The wet season does not mean that it will be rainy all day and every day. You can still enjoy days on the beautiful beaches of Zanzibar—and a bonus is that there are fewer tourists at this time of year.
Generally, these seasons are perfect for adventurous travelers, and resorts operate as usual.